convex and compact set

Asked Dec 11 '20 at 12:56

Active Dec 11 '20 at 13:01

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Hello guys I am struggling to solve this exercise could someone help me please! The question is the following:

Let $S \subseteq \mathbb{R}^k$ be a convex and compact set. For $i=1,\ldots,n$, let $f_i: S \to S$ be continuous functions. Show that there is an $x \in S$ such that $\sum_{i=1}^n f_i(x) = nx$.

edited Dec 11 '20 at 13:01

gt6989b

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asked Dec 11 '20 at 12:56

Ani

Here a Mathjax tutorial – Surb Dec 11 '20 at 12:58
2

Define $g(x) = \frac{1}{n}\sum_{i=1}^n f_i(x)$. Since $S$ is convex, $g$ maps $S$ into $S$. Also clearly $g$ is continuous. So, by Brouwer, there is some $x \in S$ with $g(x) = x$. That's what you want. – mathworker21 Dec 11 '20 at 13:07
Thanks for the help! – Ani Dec 11 '20 at 13:59

convex and compact set

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